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However, one can effectively find notations that represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's ; and given any notation for an ordinal, there is a recursively enumerable set of notations that contains one element for each smaller ordinal and is effectively ordered.
3. In set-builder notation, it is used as a separator meaning "such that"; see { : }. / 1. Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, 3 / 2 or . 2. Denotes a quotient structure.
Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets. Infinitesimals: These are smaller than any positive real number, but are nonetheless greater than zero.
Ordinal indicator – Character(s) following an ordinal number (used when writing ordinal numbers, such as a super-script) Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of ...
However, one can effectively find notations which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's ; and given any notation for an ordinal, there is a computably enumerable set of notations which contains one element for each smaller ordinal and is effectively ordered.
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [ 1 ] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
This set is ordered lexicographically with the least significant position first: we write f < g if and only if there exists x ∈ β with f(x) < g(x) and f(y) = g(y) for all y ∈ β with x < y. This is a well-ordering and hence gives an ordinal number. The definition of exponentiation can also be given by transfinite recursion on the exponent β.